Math 1303: Practice Exam 3
1. What is the meaning of the following symbols:
a) ∫ 𝑓(𝑥) 𝑑𝑥
indefinite integral or antiderivative
𝑏
b) ∫𝑎 𝑓(𝑥) 𝑑𝑥
Definite integral or area under a curve (if f(x) >= 0)
2. Find the following indefinite integrals
a) ∫ 9𝑥 2 + 4 𝑑𝑥
7
8
3
b)
∫ 3𝑥 8 + 6𝑒 𝑥 + 𝑥 − 𝑥 3 − 4 √𝑥 2 𝑑𝑥
c)
∫ (e
d)
∫(x + x
x
6
− x 2 − 5)dx
4
4
− x )dx
(this is a heavily simplified answer. The
unsimplified answer would also work)
e)
f ( x ) if f ′ ( x ) = x − 3 and f ( 4 ) = −1.
First we integrate to find f(x) =
-1 to find C, because f(4) = 2/3 * 8 – 3*4 + C = -1, so that C = 17/3
3. Evaluate the following definite integrals and simplify your answer
2
a) ∫−1 10𝑥 4 − 𝑥 𝑑𝑥
. Then we use the fact that f(4) =
𝑒
1
b) ∫1 𝑥 − 𝑥 𝑑𝑥
1
f)
∫ (t
2
− t 4 )dt
−1
e
g)
∫ (x
1
2
1
− )dx
x
4. Find the area under the graph 𝑦 = 2𝑥 − 𝑥 2 + 8 from 𝑥 = −2 to 𝑥 = 1. Make sure to sketch the function
and shade the region.
Area =
(b) Find the area under the curve y = x 2 − 4 x + 5 from x=-1 to x=3. Sketch the region.
Area =
(c) Find the area under the curve 𝑦 = 𝑒 𝑥 + 𝑥 from x = 0 to x = 1
Area =
(d) Find the area under the curve 𝑦 = 2𝑥 as x = 1 to x = 2 geometrically, then verify via integration.
5. Find the area between 𝑦 = 2𝑥 + 2 and 𝑦 = 1 − 𝑥 from x = 0 to x = 2.
Area:
(b) Find the area between 𝑓(𝑥) = 𝑥 2 − 1 and 𝑔(𝑥) = 𝑥 + 1. Make sure to sketch the functions and shade
the region.
Area =
6. Suppose the marginal cost for producing x number of widgets is C ′ = 3e x − 9q 2 − 300 , and the fixed cost is
$103. Find the formula for the cost function.
C(q) = 3e^q – 3q^3 – 300q + C, C(0) = 103 (fixed cost), so 103 = 3e^0 – 0 – 0 + C, so that C = 100
C(q) = 3e^q – 3q^3 – 300q + 100
7. After t hours of operation, a coal mine is producing coal at a rate of 40 + 2 t − 9 t 2 tons of coal per hour.
Find the formula for the output of the coal mine after t hours of operation if we know that after 2 hours, 80
tons have been mined.
Output(t) = 40t + t^2 – 3t^3 + C. We also know that Output(2) = 80, so that
40*2 + 2^2 – 3*2^3 + C = 80 so that
60 + C = 80 so that C = 20.
Thus: Output(t) = 40t + t^2 – 3t^3 + 20
8. How long will it take for $ 100 to amount to $ 1000 if invested at 6% compounded monthly? Express the
answer in years, rounded to two decimal places.
t = ln(10)/(12*ln(1.005)) = 38.47 years
9. An investor has a choice of investing a sum of money at 8% compounded semiannually or at 7.8%
compounded monthly. Which is the better of the two rates?
Effective rate of 8.0% compounded semiannually is (1 + 0.08/2)^2 – 1 = 0.0816 or 8.16%
Effective rate of 7.8% compounded monthly is (1 + 0.078/12)^12 – 1 = 0.08085 or 8.08%
Thus, the first rate is better.
9. Suppose you invest $250 at a nominal interest rate of 7% compounded quarterly.
a) What is the effective rate of interest?
b) How much is your investment worth after 5 years?
c) How would you use the effective rate to compute the answer for part (b)
a) The effective rate is (1 + 0.07/4)^4 – 1 = 0.071859 or about 7.2%
b) S = 250 * (1 + 0.07/4)^4*5 = $353.69
c) S = 250 * (1 + 0.071859)^5 = $353.69
10. Suppose $8,000 is invested in an account. How much money is in the account in 6 years if the interest rate
is 5% compounded: a) monthly b) continuously?
a) 8000*(1+0.05/12)^(6*12) = $10792.14
b) 8000*e^(0.05*6) = $10798.87 (slightly more that compounding 12 times only)
11. Suppose $700 amounted to $801.06 in a savings account after two years. If interest was compounded
quarterly, find the nominal rate of interest.
= 0.068 or 6.8%
12. A bank account pays 5.3% annual interest, compounded monthly. How much must be deposited now so that
the account contains exactly $ 12,000 at the end of one year?
=PV(0.053/12,12,0,12000) = $11,381.89
13. A trust fund for a 10-year old child is being set up with a single payment so that at age 21 the child will
receive$37,000. What is that payment if an interest rate of 6% is compounded monthly (if you use Excel for
this problem, write down the exact formula you use to obtain your answer)?
=PV(0.06/12,12*11,0,37000 = $19,154.97
14. Suppose $ 50 is placed in a savings account each month for four years. If no further deposits are made, (a)
how much is in the account after six years, and (b) how much of this amount is compound interest? Assume
that the savings account pays 6% compounded monthly (if you use Excel for this problem, write down the
exact formula you use to obtain your answer).
=FV(0.06/12,4*12,-50,0) = $2,704.89
For (b), we deposit $50 12*4=48 times, so we deposit $50 * 48 = $2,400, so the difference of $304.89 is the
amount of compound interest collected.
16. Sylvia inherited $30,000. She wants to buy a car, but also wants to put money away for graduate school. She
knows she will require $26,000 in four years for graduate school.
a) How much money should she invest now so she will have $26,000 in 4 years if the bank offers an
annual interest rate of 6.5%, compounded monthly?
b) How much money does Sylvia have left to buy her car?
=PV(0.065/12,12*4,0,26000 = $20,061.41, which will yield $26,000 in 4 years.
She therefore has $30,000 - $20.061.41 = $9,938.59 left to purchase a car now.
17. Consider the graph of a function f(x) shown below and determine whether the given quantity is
approximately positive, negative, or zero:
𝟏
f(0)>0
f ’(0)=0
f ‘’(0)<0
𝟐
f(1)=0
f ‘(1)<0
f ‘’(1)=0
𝟐
f(-1)=0
f ‘(-1)>0
f ‘’(-1)=0
a) ∫−𝟏 𝒇(𝒙) 𝒅𝒙>0
b) ∫𝟏 𝒇(𝒙) 𝒅𝒙<0
c) ∫−𝟏 𝒇(𝒙) 𝒅𝒙>0
For the following questions, you will have to use Wolfram Alpha and/or Excel. Please write down your
answers together with the formula or steps you used to produce your answer.
A. Evaluate the following expressions:
2x2
∫ 5 − x 3 dx
1
32 x 2
∫0 (5 − x) 4 dx
integrate 2x^2 / (5-x^3)
integrate 32x^2 / (5-x)^4 from 0 to 1
B. A student is saving for a vacation she is planning to take after graduation in three years. How much will she
have saved if she deposits $40 each month into an account that pays 4% compounded monthly?
=FV(0.04/12,3*12,-40,0) = $1,527.26
(To estimate: she makes 12*3 = 36 payments of $40 each, which amounts to $1,440, plus the compound
interest)
C. A parent wants her child to have $40,000 for College in 10 years. Her bank offers 4% interest, compounded
monthly, and she can afford monthly payments of $250. What will she need to do to achieve her goal?
=PV(0.04/12,12*10,-250,40000) = -$2,138.10, i.e. the parent should put up a lump sum of $2,138.10 plus
$250 per month.
(To estimate: $250 per month for 10 years amount to $250 * 12 * 10 = $30,000, not including any
compound interest. Thus, the initial investment should be less than $10,000, so our answer makes sense)